On Characterization of Non-Commutative Minkowski Space Time

Introduction

Several field theory models on k- space time have been constructed , using different techniques such as by F. Mejer and J Lukierski specially in, scalar field theory on k-Minkowski space. It has been shown by S. Tanimutra that in the flat space time, Feynman approach and minimal coupling method are equivalent which is usefull to derive the general equation of motion for a charged particle. We discuss here the applicability of Feynman approach for the case of general relativity, resulting in the derivation of the geodesic equation. We generalize the procedure for k-Minkowski space. We obtain here corrections to the geodesic equation due to the k-deformation of space time , up to the first order in the deformation parameter. WE know that a relativistic particle of mass m and electric charge e is described by in 4D-Minkowski space, where xμ(τ)  in 4D- Minkowski space, where τ is a parameter. Let us write the following relations

Where p_μ = mx + eA_μ is the canonical momentum operator, F_μ = mx, the force_μν the electromagnetic strength tensor, A_μ is a gauge field and Φ(x) is an arbitrary function odfx. We deal with gravity only, which do not require the gauge field. We only evaluate particles with no electric charge, which for a neutral particle we obtain the following equations.

Φ(x) = 0 to find the correct geodesic equation.M. Montesinos obtained that the generalization from flat to curved space by taking Eq.(1) as valid in a local Lorenz frame of reference and effect of gravity is brought in by replacing the Minkowskian metric n_μν with an arbitrary metric  g_μν (X).  He has shown that this assumption leads to geodesic equation. We choose 

Let  us assume that the ‘metric’ g_μ(x) is a function of operator X which is a symmetric tensor.

Theorem (Generalization of Commutative Space Time )

Let Xμ(τ) be a new position operator and p_μ(τ)  is  the  corresponding conjugate momenta, mX_μ = P_μ, then solution of equation may be obtain in terms of the operators given in (2). Proof

We construct operators X and P expressed as follows.

Where x_μ and p_ν  satisfy relation. By taking the derivative with respect to of Eqs.(4), we obtain the following relation.

Where p_μ = mX_μ . Combining equations (5) and (2) we obtain

Combining equations (3), (5), (6), we obtain the equation

By choosing the operator X such that is X_μ = X_μ  (x,p), LHS  of  (7)  reduces  to the equation

We now integrate Eq. (6) over p^μ  which gives the relation

By taking G_ν(x) = 0 we get

Where all Jacobi identities are satisfied. Eq. (6) is similar to the Christoffel symbol in general relativity, and Eq. (8) is similar to the geodesic equation. Let us assume that ‘metric’ is invertible and define an inverse of the symmetric tensor by the following relation

Combining the equations (4), (7) and (9) we get

Where

is really the Christoffelsymbol, we get the geodesic equation

 

Where vg_μ tensor is treated only as a symmetric tensor with an inverse defined in Eq. (10).

Theorem (Derivation of Geodric Equation In K- Minkowski Space Time)

We derive the geodesic equation for a particle moving in the non-commutative curved space time and analyze the k-Minkowski deformations of gravity. The methods taken here use k-Minkowskideformations on the flat space time. It is further generalized to k-deformed space time with arbitrary metric.

Proof

Let us define k-Minkowski space by the relation

Operators  

 

care expressed in terms of operators x and using the derivations due to J. Lukerki and H. Ruegg. We get

Where 

satisfy the identity 

Let us solve Eq. (11) up to the first order in deformation parameter a, which gives

Where parameters of the realization a, b, and g satisfy the a constraint

Let  us  define an  operator 

which commutes with

, i.e.

Any function of ˆy also commutes with

Where we take only first order in the modified form of give 

and 

up to the first order in

The canonical momentum operator ( in e = 0 case )

as derived by the relation

Hence , we conclude that construction due to E. HariKumar via Feynman approach satisfy all Jacobi identities. The condition that comes by taking the derivative of Eq. (12) with respect to τ is

Hence the geodesic equation in the k- Minkowski space time is established.

Theorem ( k-Dependent Corrections to the Geodesic Equation )

Proof

Let us choose the conjugate pairs (x , p ) and for flat non-commutative space time, 

in commutative and non- commutative space respectively. It is shown that all the operators in the flat non-commutative space time are expressed in terms of x , p and deformation parameter a. For the non-commutative space time with curvature. Let us construct it as functions of x , p and deformation parameter . In case of neutral particles , conjugate momenta is  given by  

We derive the corrections to the geodesic equation due to the k- deformation of space time. Let us consider the relation.

Where

And

satisfies Eq. (12).

satisfy all the Jacobi Indentifies and the relation 

But in the limit a ®0, it implies that

By taking limit 

we get the following relations

Combining   these  equation, and comparing it with (13) & (14)

substitute

 

with a function that commutes with

 

That is with

 

Thus , we obtain the required form 

By an appropriate application of (14), we find that this construction satisfioes all Jacobi identities and Eq. (15) is also satisfied to all orders in a

Hence , get its modified form as follows 

 

An alternative construction of the operator up to the first order in the deformation parameter may be obtained by differentiating Eq. (17) with respect to τ.

It satisfies antisymmetric part of

 

We write it as follows

 

Where

 

and

Combining Eq. (16) and (17), We get

 

By taking the limit a a →0, we obtain

 

Hence , the case of up to the first order in the deformation parameter a

i.e.

Here 

and we get the following relations

We get constraints on 

that the Jacobi identities must be satisfied up to the first order in a.

 

We get

 

Let us construct

 

from

 

and

and express it as

Hence

is determined by  the parameters a and b and four

 

More free parameters. This general used form but valid up to the first order in the deformation

parameter a. The construction

 

where is the special case of this general procedure.

Conclusion

The principal characteristic of this approach is that all the corrections depend on the choice of realization of the parameters on the mass of the test particle.

 

We analysed here the a-dependent correction to the Newtonian limit of the geodesic equation which shows that the Newtonian force/potential remains radial, but depends on the mass of the test particle.

 

We have shown here that the κ-deformed commutation relations between phase space  variables induce modified incertainty relations 

 

It has been shown that the commutative limit

  gives rise to the metric

 and by analogy, we interpret 

    

 non- commutative metric  .

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